MinT - Best Known (24, 53, s)-Nets in Base 128

Best Known (24, 53, s)-Nets in Base 128

(24, 53, 438)-Net over F₁₂₈ — Constructive and digital

Digital (24, 53, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (9, 38, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(24, 53, 517)-Net in Base 128 — Constructive

(24, 53, 517)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 16, 257)-net in base 128, using
  - base change [i] based on digital (0, 14, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
2. (8, 37, 260)-net in base 128, using
  - 3 times m-reduction [i] based on (8, 40, 260)-net in base 128, using
    - base change [i] based on digital (3, 35, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(24, 53, 881)-Net over F₁₂₈ — Digital

Digital (24, 53, 881)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(24, 53, 3194721)-Net in Base 128 — Upper bound on s

There is no (24, 53, 3194722)-net in base 128, because

1 times m-reduction [i] would yield (24, 52, 3194722)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 37 576760 465840 412916 106161 265044 052056 086963 367146 532731 371411 394376 574435 670444 260784 265913 455729 539436 562512 > 128⁵² [i]